Physics 207, Lecture 12, Oct. 15

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Physics 207, Lecture 12, Oct. 15

• Agenda Finish Chapter 9, start Chapter 10

• Chapter 9 Momentum Impulse
• Collisions
• Momentum conservation in 2D
• Impulse

• Assignment
• HW5 due Wednesday
• HW6 posted soon

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Impulse Linear Momentum

• Transition from forces to conservation laws
• Newtons Laws ? Conservation Laws
• Conservation Laws ? Newtons Laws
• They are different faces of the same physics
  phenomenon for special cases

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Lecture 12, Exercise 1Momentum Conservation

• Two balls of equal mass are thrown horizontally
  with the same initial velocity. They hit
  identical stationary boxes resting on a
  frictionless horizontal surface.
• The ball hitting box 1 bounces elastically back,
  while the ball hitting box 2 sticks.
• In which case does the box ends up moving
  fastest ?
• No external force then, vectorially, COM

1. Box 1
2. Box 2
3. Same

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Lecture 12, Exercise 1Momentum Conservation

• Which box ends up moving fastest ?
• Examine the change in the momentum of the ball.
• In the case of box 1 the balls momentum changes
  sign and so its net change is largest. Since
  momentum is conserved the box must have the
  largest velocity to compensate.

(A) Box 1 (B) Box 2 (C)
same
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1
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A perfectly inelastic collision in 2-D

• Consider a collision in 2-D (cars crashing at a
  slippery intersection...no friction).

V
v1
m1 m2
m1
m2
v2
before
after

• If no external force momentum is conserved.
• Momentum is a vector so px, py and pz

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Elastic Collisions

• Elastic means that the objects do not stick.
• There are many more possible outcomes but, if no
  external force, then momentum will always be
  conserved
• Start with a 1-D problem.

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Elastic Collision in 1-D
m2
m1
before
v2b
v1b
x
m2
m1
after
v2a
v1a
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Force and Impulse (A variable force applied for
a given time)

• Gravity usually a constant force to an object
• Springs often provide a linear force (-k x)
  towards its equilibrium position (Chapter 10)
• Collisions often involve a varying force
• F(t) 0 ? maximum ? 0
• We can plot force vs time for a typical
  collision. The impulse, J, of the force is a
  vector defined as the integral of the force
  during the time of the collision.

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Force and Impulse (A variable force applied for
a given time)

• J reflects momentum transfer

F
Impulse J area under this curve ! (Transfer of
momentum !)
Impulse has units of Newton-seconds
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Force and Impulse

• Two different collisions can have the same
  impulse since J depends only on the momentum
  transfer, NOT the nature of the collision.

same area
F
t
?t
?t
?t big, F small
?t small, F big
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Average Force and Impulse
Fav
F
Fav
t
?t
?t
?t big, Fav small
?t small, Fav big
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Example from last time

• A 2 kg cart initially at rest on frictionless
  horizontal surface is acted on by a 10 N
  horizontal force along the positive x-axis for 2
  seconds what is the final velocity?
• F is in the x-direction F ma so a F/m
  5 m/s2
• v v0 a t 0 m/s 2 x 5 m/s 10 m/s
  (x-direction)
• but mv F t which is the area with respect
  to F(t) curve

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Lecture 12, Exercise 2Force Impulse

• Two boxes, one heavier than the other, are
  initially at rest on a horizontal frictionless
  surface. The same constant force F acts on each
  one for exactly 1 second.
• Which box has the most momentum after the force
  acts ?

1. heavier
2. lighter
3. same
4. cant tell

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Lecture 12, Exercise 2Force Impulse

• Two boxes, one heavier than the other, are
  initially at rest on a horizontal frictionless
  surface. The same constant force F acts on each
  one for exactly 1 second.
• Which box has the most momentum after the
  force acts ?

(A) heavier (B) lighter
(C) same
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Boxers
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• Back of the envelope calculation
• (1) marm 7 kg (2) varm7 m/s (3) Impact
  time ?t 0.01 s
•  
• ? Impulse J ?p marm varm 49 kg m/s
•  
• ? F J/?t 4900 N
•  
• (1) mhead 6 kg
•  
• ? ahead F / mhead 800 m/s2 80 g !
•  
• Enough to cause unconsciousness 40 of fatal
  blow

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Woodpeckers

• During "collision" with a tree 
• ahead 600 - 1500 g
•  
• How do they survive?
•  
• Jaw muscles act as shock absorbers
• Straight head trajectory reduces damaging
  rotations (rotational motion is very problematic)

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Chapter 10 Energy

• What do we mean by an isolated system ?
• What do we mean by a conservative force ?
• If a force acting on an object act for a period
  of time then we have an Impulse ? change
  (transfer) of momentum
• What if we consider this force acting over a
  distance
• Can we identify another useful quantity?

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Energy

• Fy m ay and let the force be constant
• y(t) y0 vy0 t ½ ay t2 ?Dy y(t)-y0 vy0 t
  ½ ay t2
• vy (t) vy0 ay t ? t (vy -
  vy0) / ay Dvy / ay
• So Dy vy0 Dvy / ay ½ ay
  (Dvy/ay)2
• (vy vy0 - vy02 ) / ay ½ (vy2 - 2vy
  vy0vy02 ) / ay
• 2 ay Dy (vy2 - vy02 )
• Finally may Dy ½ m (vy2 - vy02 )
• If falling -mg Dy ½ m (vy2 - vy02 )

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Energy

• -mg Dy ½ m (vy2 - vy02 )
• -mg (yf yi) ½ m ( vyf2 -vyi2 )

A relationship between y displacement and y speed
Rearranging ½ m vyi2 mgyi ½ m vyf2 mgyf
We associate mgy with the gravitational
potential energy
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Energy

• Notice that if we only consider gravity as the
  external force then
• then the x and z velocities remain constant
• To ½ m vyi2 mgyi ½ m vyf2 mgyf
• Add ½ m vxi2 ½ m vzi2 and ½ m vxf2
  ½ m vzf2
• ½ m vi2 mgyi ½ m vf2 mgyf
• where vi2 vxi2 vyi2 vzi2
• ½ m v2 terms are referred to as kinetic
  energy

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Energy

• If only conservative forces are present, the
  total energy (sum of potential, U, and kinetic
  energies, K) of a system is conserved.

Emech K U
Emech K U constant

• K and U may change, but E K Umech remains a
  fixed value.

Emech is called mechanical energy
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Another example of a conservative system The
simple pendulum.

• Suppose we release a mass m from rest a distance
  h1 above its lowest possible point.
• What is the maximum speed of the mass and where
  does this happen ?
• To what height h2 does it rise on the other side ?

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Physics 207, Lecture 12, Oct. 15

• Agenda Finish Chapter 9, start Chapter 10

• Chapter 9 Momentum
• Collisions
• Momentum conservation in 2D
• Impulse

• Assignment
• HW5 due Wednesday
• HW6 posted soon
• Finish Chapter 10, Start 11 (Work)

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Example of 2-D Elastic collisionsBilliards

• If all we are given is the initial velocity of
  the cue ball, we dont have enough information to
  solve for the exact paths after the collision.
  But we can learn some useful things...

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Billiards

• Consider the case where one ball is initially at
  rest.

after
before
pa q
pb
vcm
Pa f
F
The final direction of the red ball will depend
on where the balls hit.
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Billiards All that really matters is
conservation of energy and momentum

• COE ½ m vb2 ½ m va2 ½ m Va2
• x-dir COM m vb m va cos q m Vb cos f
• y-dir COM 0 m va sin q m Vb sin f

• The final directions are separated by 90 q
  f 90